Final answer:
The equation of the parabola is found by using the x-intercepts to set up a factored form, inserting a known point to solve for the variable 'a', and then substituting back to get the final equation.
Step-by-step explanation:
The equation of a parabola that has its x-intercepts at (1+sqrt(5),0) and (1-sqrt(5),0) can be written in vertex form or factored form. Since we have the x-intercepts, let's use the factored form of the quadratic equation, y = a(x - r1)(x - r2), where r1 and r2 are the roots of the parabola.
Using the given x-intercepts, the equation becomes y = a(x - (1 + sqrt(5)))(x - (1 - sqrt(5))). To find the value of 'a', we can use the point through which it passes, which is (4,8). Substituting the point (4,8) into the equation gives us:
8 = a(4 - (1 + sqrt(5)))(4 - (1 - sqrt(5)))
Solving for 'a', we first simplify inside the parentheses and then multiply the expressions, followed by dividing both sides by the resulting coefficient of 'a'. After finding the value of 'a', we can plug it back into our equation to obtain the final form of the parabola.